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Macaulay2Doc :: isHomogeneous

isHomogeneous -- whether something is homogeneous (graded)

Synopsis

Description

i1 : isHomogeneous(ZZ)

o1 = true
i2 : isHomogeneous(ZZ[x,y])

o2 = true
i3 : isHomogeneous(ZZ[x,y]/(x^3-x^2*y+3*y^3))

o3 = true
i4 : isHomogeneous(ZZ[x,y]/(x^3-y-3))

o4 = false

Quotients of multigraded rings are homogeneous, if the ideal is homogeneous.

i5 : R = QQ[a,b,c,Degrees=>{{1,1},{1,0},{0,1}}];
i6 : I = ideal(a-b*c);

o6 : Ideal of R
i7 : isHomogeneous I

o7 = true
i8 : isHomogeneous(R/I)

o8 = true
i9 : isHomogeneous(R/(a-b))

o9 = false

Polynomial rings over polynomial rings are multigraded.

i10 : A = QQ[a]

o10 = A

o10 : PolynomialRing
i11 : B = A[x]

o11 = B

o11 : PolynomialRing
i12 : degree x

o12 = {1, 0}

o12 : List
i13 : degree a_B

o13 = {0, 1}

o13 : List
i14 : isHomogeneous B

o14 = true

A matrix is homogeneous if each entry is homogeneous of such a degree that the matrix has a well-defined degree.

i15 : S = QQ[a,b];
i16 : F = S^{-1,2}

       2
o16 = S

o16 : S-module, free, degrees {1, -2}
i17 : isHomogeneous F

o17 = true
i18 : G = S^{1,2}

       2
o18 = S

o18 : S-module, free, degrees {-1, -2}
i19 : phi = random(G,F)

o19 = {-1} | 8a2+ab+3b2        0 |
      {-2} | 7a3+8a2b+3ab2+3b3 7 |

              2       2
o19 : Matrix S  <--- S
i20 : isHomogeneous phi

o20 = true
i21 : degree phi

o21 = {0}

o21 : List

Modules are homogeneous if their generator and relation matrices are homogeneous.

i22 : M = coker phi

o22 = cokernel {-1} | 8a2+ab+3b2        0 |
               {-2} | 7a3+8a2b+3ab2+3b3 7 |

                             2
o22 : S-module, quotient of S
i23 : isHomogeneous(a*M)

o23 = true
i24 : isHomogeneous((a+1)*M)

o24 = false

Note that no implicit simplification is done. Consider the following cautionary example.

i25 : R = QQ[x]

o25 = R

o25 : PolynomialRing
i26 : isHomogeneous ideal(x+x^2, x^2)

o26 = false

For developers: isHomogeneous also has a method for EngineRings.

Caveat

No computation on the generators and relations is performed. For example, if inhomogeneous generators of a homogeneous ideal are given, then the return value is false.

See also

Ways to use isHomogeneous :